Abstract: By introducing some special functions to characterize the constant factor, a Hardy-Hilbert type integral inequality with the kernel $\ln(1+\mathrm{e}^{-\alpha x^{\lambda_1}y^{\lambda_2}})$ is obtained, and its equivalent form is considered. The constant factors of the equivalent inequalities are proved being the best possible.

Key words: Hilbert-type integral inequality, \quad weight function, the best constant factor